Uniform phase modulation via control of refractive index in a thermal atom vapor with vanishing diffraction or absorption

A scheme is proposed to achieve substantial controllable phase modulation for a probe field propagating through a thermal atomic vapor in double-$\Lambda$ configuration. The phase modulation is based on manipulating the linear susceptibility of the probe field, paraxial diffraction is eliminated by exploiting the thermal motion of atoms, and residual absorption is compensated via an incoherent pump field. As a result of the homogeneous change in the refractive index, a strong controllable uniform phase modulation without paraxial diffraction is achieved essentially independent of the spatial profile or the intensity of the probe field. This phase shift can be controlled via the intensities of the control or the incoherent pump fields. A possible proof-of-principle experiment in alkali-metal atoms is discussed.

Figure 1

Schematic illustration for phase modulation acquired by a laser field throughout propagation in different media. The arrows indicate the phase distribution of the laser field. (a) In free space, diffraction leads to spatial spreading and a small nonuniform phase distribution in the transverse plane. (b) In a regular dispersive medium, spatial spreading and nonuniform phase distortion due to diffraction occur. (c) In the setup discussed here, a strong spatially uniform phase shift is achieved, together with cancellation of paraxial diffraction.

Figure 2

(a) Atomic level scheme. The four-level double-$\Lambda$ system interacts with the copropagating probe and control fields and a two-way incoherent pump field. The field configuration is sketched in (b). Due to the atomic motion, the paraxial diffraction can be eliminated.

Figure 3

(a) Power and spatial width of the Gaussian probe beam as a function of the propagation distance. Note the small plot range of only a few percent relative change, suggesting that the probe field power and width remain approximately unchanged. (b) The accumulated phase shift as a function of propagation distance. $\pi$-phase flips can be achieved already in a small fraction of the Rayleigh length $z_R$. Parameters are $n_0=1.5\times {10}^{12}{\text{cm}}^{-3},{\lambda }_p=795\text{nm},T=300\text{K},v_{\text{th}}=240\text{m}/\mathrm{s},\Delta k=22.8{\text{m}}^{-1},{\gamma }_c=2000\Delta k{v}_{\text{th}},\Gamma =2\pi \times 5.75\text{MHz},{\Gamma }_{31}=\Gamma /4,{\Gamma }_{32}=\Gamma /6,{\Gamma }_{41}=\Gamma /12,{\Gamma }_{42}=\Gamma /2,{\gamma }_{21}=0.001{\Gamma }_{31},{\Omega }_c=1.4{\Gamma }_{31},p=0.65{\Gamma }_{31}$.

Figure 4

(a) Phase shift of the transmitted Gaussian probe field against the intensity of the control after propagating one Rayleigh length in the thermal atomic gas. (b) The power and width of the outgoing probe field as a function of the control. Other parameters are as in Fig. 3.

Figure 5

(a) Phase shift of the output Gaussian probe field as a function of the incoherent pump rate after propagating one Rayleigh length in the thermal atomic gas. (b) The power and width of the outgoing probe field functions of the incoherent pump rate. Other parameters are as in Fig. 3.

Figure 6

(a) Phase shift (in units of $\pi$) of the output Gaussian probe field plotted against the intensities of the control and incoherent pump fields after propagating one Rayleigh length in the thermal atomic gas. particular phase modulations can be achieved in various ways by tuning the control and incoherent pump simultaneously. (b) and (c) show the corresponding relative power $\left(\text{ln}[P_{\text{out}}/P_{\text{in}}]\right)$ and width $\left(w_{\text{out}}/w_p\right)$ of the outgoing probe field. Other parameters are the same as in Fig. 3.

Figure 7

Spatial phase variation of the probe field in the transverse plane in units of $\pi$ after a propagation distance $z_f=0.168z_R$. Results are shown for free space (a), a cold EIT medium (b), and the thermal vapor considered here (c). The parameters chosen for the cold EIT medium are the same as in Fig. 3 except for $T=0\text{K},{\gamma }_c=0,{\gamma }_{21}=0,p=0,{\Delta }_p=-0.001{\Gamma }_{31}$. In the thermal vapor, the probe acquires an almost uniform phase $\pi$, as compared to a small nonuniform phase gained in the cold EIT medium or in free space. (d)–(f) Phase difference, which is defined as $(\varphi [x=y=0]-\varphi [x=y=w_p])/(\varphi [x=y=0]+\varphi [x=y=w_p])$, as a function of propagation distance for the three cases. The starting points for $z$ in (d)–(f) are $z=0.001z_R$, since at $z=0$ the phase difference for all cases should be zero. Other parameters are the same as in Fig. 2.